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Mirrors > Home > MPE Home > Th. List > Mathboxes > ineccnvmo | Structured version Visualization version GIF version |
Description: Equivalence of a double restricted universal quantification and a restricted "at most one" inside a universal quantification. (Contributed by Peter Mazsa, 2-Sep-2021.) |
Ref | Expression |
---|---|
ineccnvmo | ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5807 | . . 3 ⊢ Rel ◡𝐹 | |
2 | id 22 | . . . 4 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧) | |
3 | 2 | inecmo 35061 | . . 3 ⊢ (Rel ◡𝐹 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥)) |
4 | 1, 3 | ax-mp 5 | . 2 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥) |
5 | brcnvg 5600 | . . . . 5 ⊢ ((𝑦 ∈ V ∧ 𝑥 ∈ V) → (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦)) | |
6 | 5 | el2v 3422 | . . . 4 ⊢ (𝑦◡𝐹𝑥 ↔ 𝑥𝐹𝑦) |
7 | 6 | rmobii 3336 | . . 3 ⊢ (∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
8 | 7 | albii 1782 | . 2 ⊢ (∀𝑥∃*𝑦 ∈ 𝐵 𝑦◡𝐹𝑥 ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
9 | 4, 8 | bitri 267 | 1 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑦 = 𝑧 ∨ ([𝑦]◡𝐹 ∩ [𝑧]◡𝐹) = ∅) ↔ ∀𝑥∃*𝑦 ∈ 𝐵 𝑥𝐹𝑦) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∨ wo 833 ∀wal 1505 = wceq 1507 ∀wral 3088 ∃*wrmo 3091 Vcvv 3415 ∩ cin 3828 ∅c0 4178 class class class wbr 4929 ◡ccnv 5406 Rel wrel 5412 [cec 8087 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-xp 5413 df-rel 5414 df-cnv 5415 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-ec 8091 |
This theorem is referenced by: ineccnvmo2 35066 |
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